{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/legendre-memory-units-continuous-time","title":"Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks","arxiv_id":null,"date":"2019-12-01","proceeding":"NeurIPS 2019 12","authors":["Aaron Voelker","Ivana Kajić","Chris Eliasmith"],"abstract":"We propose a novel memory cell for recurrent neural networks that dynamically maintains information across long windows of time using relatively few resources. The Legendre Memory Unit~(LMU) is mathematically derived to orthogonalize its continuous-time history -- doing so by solving $d$ coupled ordinary differential equations~(ODEs), whose phase space linearly maps onto sliding windows of time via the Legendre polynomials up to degree $d - 1$. Backpropagation across LMUs outperforms equivalently-sized LSTMs on a chaotic time-series prediction task, improves memory capacity by two orders of magnitude, and significantly reduces training and inference times. LMUs can efficiently handle temporal dependencies spanning $100\\text{,}000$ time-steps, converge rapidly, and use few internal state-variables to learn complex functions spanning long windows of time -- exceeding state-of-the-art performance among RNNs on permuted sequential MNIST. These results are due to the network's disposition to learn scale-invariant features independently of step size. Backpropagation through the ODE solver allows each layer to adapt its internal time-step, enabling the network to learn task-relevant time-scales. We demonstrate that LMU memory cells can be implemented using $m$ recurrently-connected Poisson spiking neurons, $\\mathcal{O}( m )$ time and memory, with error scaling as $\\mathcal{O}( d / \\sqrt{m} )$. We discuss implementations of LMUs on analog and digital neuromorphic hardware.","url_abs":"http://papers.nips.cc/paper/9689-legendre-memory-units-continuous-time-representation-in-recurrent-neural-networks","url_pdf":"http://papers.nips.cc/paper/9689-legendre-memory-units-continuous-time-representation-in-recurrent-neural-networks.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"legendre-memory-units-continuous-time","repo_url":"https://github.com/abr/neurips2019","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"tf","reach":null},{"paper_slug":"legendre-memory-units-continuous-time","repo_url":"https://github.com/hrshtv/pytorch-lmu","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"pytorch","reach":{"status":"unanswered"}}],"tasks":[{"task_slug":"sequential-image-classification","task_name":"Sequential Image Classification"},{"task_slug":"time-series-1","task_name":"Time Series"},{"task_slug":"time-series","task_name":"Time Series Analysis"},{"task_slug":"time-series-prediction","task_name":"Time Series Prediction"}],"methods":[{"method_slug":"lmu","method_name":"LMU"}],"datasets_introduced":[],"methods_introduced":[{"slug":"lmu","name":"LMU","full_name":"Legendre Memory Unit"}],"results":[{"leaderboard":"/sota/sequential-image-classification-on-sequential","task":"Sequential Image Classification","dataset":"Sequential MNIST","model":"LMU","rank_in_archive_order":14,"of":30,"metrics":{"Permuted Accuracy":"97.2%"},"uses_additional_data":false}],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}